2/ Sampling at the Nyquist rate
In this example fs = 2W, i.e., there are two samples per period of the cosine wave. The cosine wave is sampled at the Nyquist rate.
If the sampled signal is passed through an ideal lowpass filter with cut-off frequency just above W, which is the frequency of the cosine wave, then the original cosine wave is again recovered completely from its samples.
In this example fs = 2W, but a sine wave rather than a cosine wave is sampled at the Nyquist rate, i.e., x(t) = sin(2πWt).
The sinusoid is sampled at the zero crossings — both the time function and its Fourier transform are zero. The sinusoid cannot be recovered from its samples.
Thus, sampling exactly at the Nyquist rate does not always lead to recovery of the original signal, and recovery depends upon the phase of the sinusoid. To understand this, we sample x(t)=Acos(2πWt+θ) at the Nyquist rate.
For x(t) = Acos(2πWt + θ), y(t) = (Acos θ) cos(2πWt). Thus, there is an ambiguity in the amplitude and the original phase of the cosinusoid is lost.
In general, sampling at, as opposed to above, the Nyquist rate will not lead to recovery of the original signal from its samples.
3/ Sampling below the Nyquist rate — undersampling
In this example fs = (3/2)W and x(t) = sin(2πWt).
If the sampled signal is passed through an ideal lowpass filter with cut-off frequency just above W, then y(t) = sin(2πWt) − sin(2π(W/2)t). The second term is the sine wave aliased to the frequency of W/2.
The time waveforms when undersampling with fs = (3/2)W and with an ideal lowpass filter with cut-off frequency just above W results in the waveforms shown below. x(t) = sin(2πWt) and y(t) = sin(2πWt) − sin(2π(W/2)t).
If the sampled signal is passed through an ideal lowpass filter with cut-off frequency just above W/2, then y(t) = −sin(2π(W/2)t), i.e., the undersampled sine wave has been reduced in frequency from W to W/2.
The original signal, x(t) = sin(2πWt), and the sampled and filtered signal, y(t) = −sin(2π(W/2)t), are compared below.
The dark points show the locations of samples of x(t) = sin(2πWt) which is sampled at fs = (3/2)W. The sampled signal is passed through an ideal lowpass filter whose cut-off frequency is just above W/2 to yield y(t) = −sin(2π(W/2)t).
VII. DEMONSTRATIONS
1/ The effect of sampling on sinusoidal audio signals
2/ The effect of sampling and quantization on audio signals
3/ Quantization
To transfer a CT signal into a computer, the CT signal must be sampled and quantized. Quantization converts a sample whose amplitude is specified with infinite precision into a number with limited precision. The transfer function of the quantizer is shown for quantizers of different precision specified by the number of bits. The A/D and D/A converter in this demonstration has 14 bit precision.
The difference between the original signal and the quantized signal constitutes an error. The error decreases as the number of quantization levels is increased.
4/ Stroboscopic illumination of a fan
The sampled time function appears as if θ(t) is increasing so that motion of the fan appears clockwise.
The sampled time function appears as if θ(t) is decreasing so that motion of the fan appears counterclockwise.
5/ Effect of sampling on images
We examine sampling of images using a MATLAB software package that allows display of images, sampled images, reconstructed images both with and without anti-alias filtering.
The image on the left has been sampled by keeping every 4th pixel to produce the sampled image on the right.
The sampled image has been reconstructed with a zero-order hold (staircase approximation) on the left and a first-order hold (linear interpolation) on the right.
The original image is shown on the left and the same image passed through an anti-alias filter appropriate to sampling every 4th point is shown on the right.
The effect of the anti-aliasing filter is seen by comparing the reconstructed filter without anti-alias filtering (left) with that using an anti-alias filter (right) both using a first-order hold (linear interpolation).
VI. CONCLUSIONS
The central idea in sampling a CT signal is the Sampling Theorem and its consequences.
- Let x(t) be a bandlimited time function, i.e., X(f) = 0 for |f|>|W|. Let x(t) be sampled at a sampling frequency fs>2W, where 2W is called the Nyquist rate. Then x(t) can, in principle, be completely recovered from the samples.
- x(t) can be recovered bypassing the sampled signal through an ideal LPF with cutoff frequency at f = W.
- The recovery of x(t) is equivalent to interpolating the samples of x(t) with a sinc function which is the inverse Fourier transform of the ideal LPF.